scholarly journals A construction of complex analytic elliptic cohomology from double free loop spaces

2021 ◽  
Vol 157 (8) ◽  
pp. 1853-1897
Author(s):  
Matthew Spong
Keyword(s):  
Elliptic Curves ◽  
Inverse Limit ◽  
Loop Space ◽  
Equivariant Cohomology ◽  
Cohomology Theory ◽  
Free Loop Space ◽  
Loop Spaces ◽  
Elliptic Cohomology ◽  
Finite Dimensional ◽  
Moduli Stack

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$ , the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$ .

scholarly journals On the geometry of free loop spaces

2002 ◽  
Vol 30 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. Manoharan
Keyword(s):  
Fundamental Form ◽  
Open Subset ◽  
Loop Space ◽  
Free Loop Space ◽  
Loop Spaces ◽  
Central Extensions ◽  
Christoffel Symbols

We verify the following three basic results on the free loop spaceLM. (1) We show that the set of all points, where the fundamental form onLMis nondegenerate, is an open subset. (2) The connections of a Fréchet bundle overLMcan be extended toS1-central extensions and, in particular, there exist natural connections on the string structures. (3) The notion of Christoffel symbols and the curvature are introduced onLMand they are described in terms of Christoffel symbols ofM.

scholarly journals On the Borel cohomology of free loop spaces

2003 ◽  
Vol 93 (2) ◽  
pp. 185 ◽  
Author(s):  
Iver Ottesen
Keyword(s):  
Loop Space ◽  
Free Loop Space ◽  
Loop Spaces ◽  
Type K

Let $X$ be a space and let $K = H^*(X; \boldsymbol F_p)$ where $p$ is an odd prime. We construct functors $\overline \Omega$ and $\ell$ which approximate cohomology of the free loop space $\Lambda X$ as follows: There are homomorphisms $\overline \Omega(K) \to H^*(\Lambda X; \boldsymbol F_p)$ and $\ell(K)\to H^*(E\boldsymbol T\times_T\Lambda X;\boldsymbol F_p)$. These are isomorphisms when $X$ is a product of Eilenberg-MacLane spaces of type $K(\boldsymbol F_p,n)$ for $n \geq 1$.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

scholarly journals Hodge decomposition of string topology

2021 ◽  
Vol 9 ◽  
Author(s):  
Yuri Berest ◽  
Ajay C. Ramadoss ◽  
Yining Zhang
Keyword(s):  
Lie Algebras ◽  
General Theorem ◽  
Hodge Decomposition ◽  
String Topology ◽  
Free Loop Space ◽  
Oriented Manifold ◽  
Universal Enveloping Algebras ◽  
Finite Dimensional ◽  
Equivariant Homology ◽  
Simply Connected

Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals Morava $K$-theory and the free loop space

1992 ◽  
Vol 114 (1) ◽  
pp. 243-243
Author(s):  
John McCleary ◽  
Dennis A. McLaughlin
Keyword(s):  
Loop Space ◽  
Free Loop Space ◽  
K Theory

scholarly journals A new description of equivariant cohomology for totally disconnected groups

2008 ◽  
Vol 1 (3) ◽  
pp. 431-472 ◽  
Author(s):  
Christian Voigt
Keyword(s):  
Equivariant Cohomology ◽  
Simplicial Complexes ◽  
Cyclic Homology ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Natural Class ◽  
New Approach ◽  
Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

scholarly journals On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$

2017 ◽  
Vol 121 (2) ◽  
pp. 186
Author(s):  
Iver Ottosen
Keyword(s):  
Loop Space ◽  
Vector Bundles ◽  
Morse Decomposition ◽  
Energy Integral ◽  
Chern Classes ◽  
Free Loop Space ◽  
Stiefel Manifolds ◽  
Mod P

We give a description of the negative bundles for the energy integral on the free loop space $L\mathbb{C}\mathrm{P}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.

Sign in / Sign up

Export Citation Format

Share Document